How does gravity save or kill Q-balls?

We explore stability of gravitating Q-balls with potential $V_4(\varphi )=\frac{m^2}{2}{\varphi }^2-\lambda {\varphi }^4+\frac{{\varphi }^6}{M^2}$ via catastrophe theory, as an extension of our previous work on Q-balls with potential $V_3(\varphi )=\frac{m^2}{2}{\varphi }^2-\mu {\varphi }^3+\lambda {\varphi }^4$. In flat spacetime Q-balls with $V_4$ in the thick-wall limit are unstable and there is a minimum charge $Q_{\mathrm{m}\mathrm{i}\mathrm{n}}$, where Q-balls with $Q<Q_{\mathrm{m}\mathrm{i}\mathrm{n}}$ are nonexistent. If we take self-gravity into account, on the other hand, there exist stable Q-balls with arbitrarily small charge, no matter how weak gravity is. That is, gravity saves Q-balls with small charge. We also show how stability of Q-balls changes as gravity becomes strong.